Binocular stereopsis is the ability to use differences between the images presented to the two eyes (binocular disparities) to perceive the three dimensional structure of the outside world. The geometry of binocular vision ensures that these differences take a particular form: an image on one part of the right retina can be almost exactly matched to an equivalent image region on the left retina by a horizontal translation. Understanding how neurons are able to signal these disparities is an excellent model system for studying how neuronal processing generates useful perceptual representations. [unreadable] [unreadable] Current explanations for this mechanism rely upon linear processing of the monocular images. We show that this results in neuronal responses that devote part of their dynamic range to impossible stimuli. In order to determine whether cortical neurons have this unexpected property, we conducted two studies in which sinusoidal luminance gratings were summed. We manipulated the interocular phase difference of each grating component independently. In a stimulus with just two components, we were able to explore all possible combinations. Within this space of combinations, natural disparities (pure translation) all fall along a single line. We found that approximately half of the neurons we studied in the striate cortex did indeed show maximal responses for unnatural combinations of interocular phase. The other half, however, all showed maximal responses to natural stimuli, in a way that current models do not explain. We developed a modification of the binocular energy model that was able to explain the experimental data. One attraction of the new model is that it could also be applied to the outputs of the striate cortex, and hence might explain some of the transformations in binocular signals that occur in extrastriate cortex. [unreadable] [unreadable] The new model makes a distinctive prediction about responses to stimuli with many sinusoidal components, if the interocular phase differences are manipulated in a particular fashion. We recorded the responses of neurons in the striate cortex, and in the first extrastriate area (V2) to this manipulation. Responses in both areas confirmed the predications of the model. Furthermore, quantitative analysis of the responses from V2 neurons indicated that they did not simply inherit this property from disparity selective neurons in striate cortex. Rather, the mechanism proposed in our model seems to operate in the projection from V1 to V2.